In the 19th century the Eisenlohr projection was proposed: a map of the whole earth on the plane, that is conformal everywhere and (I hear it said) has no isolated points at which it is not conformal. Cartographers seem to consider it a "novelty projection" for which there is no market.
So: Is there a one-to-one conformal mapping from $\mathbb C\cup\{\infty\}$ into a bounded subset of $\mathbb C$, with no isolated points of non-conformality? And is this an example of that?
https://books.google.com/books?id=Z7MufM7e5xMC&pg=PA184&lpg#v=onepage&q&f=false